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Theorem elprg 3617
Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15, generalized. (Contributed by NM, 13-Sep-1995.)
Assertion
Ref Expression
elprg  |-  ( A  e.  V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )

Proof of Theorem elprg
StepHypRef Expression
1 eqeq1 2262 . . 3  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
2 eqeq1 2262 . . 3  |-  ( x  =  A  ->  (
x  =  C  <->  A  =  C ) )
31, 2orbi12d 693 . 2  |-  ( x  =  A  ->  (
( x  =  B  \/  x  =  C )  <->  ( A  =  B  \/  A  =  C ) ) )
4 dfpr2 3616 . 2  |-  { B ,  C }  =  {
x  |  ( x  =  B  \/  x  =  C ) }
53, 4elab2g 2884 1  |-  ( A  e.  V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    \/ wo 359    = wceq 1619    e. wcel 1621   {cpr 3601
This theorem is referenced by:  elpr  3618  elpr2  3619  elpri  3620  eltpg  3636  ifpr  3641  prid1g  3692  ordunpr  4575  cnsubrg  16380  atandm  20120  eupath2lem1  23259  repfuntw  24513
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2759  df-un 3118  df-sn 3606  df-pr 3607
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